Fourier Analysis is the decomposition of any square-integrable functions into an infinite series of trigonometric functions. Here we show that the trigonometric polynomial functions are dense in the space of periodic continuous functions, and thus can be used as good approximations. This implies...

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## A Tutorial in Data Science: Lecture 8 – The Functional Theory of Communication in Stochastic Processes

by Justin Petrillo | Math Lecture

From Laplace's Calculus of Probability (Lecture 3), a finite difference equation explains probabilistically how a game evolves over discrete time, defining thus a stochastic process and specifically a Markov Chain where there is finite recursion or limited historical dependence. While one can...

## A Tutorial in Data Science: Lecture 9 – The Functional Theory of Communication in Dynamic Systems

by Justin Petrillo | Math Lecture

We consider thus the functional notion of communication within the theory of dynamic systems, where the finite difference equation, or flow of a differential equation, is iterated upon as a function. Within this framework, there is a true underlying deterministic system which can be explained by a...

## A Tutorial in Data Science: Lecture 10 – The Fundamental Law of Data Science (Inter-Communication is Functional Chaos)

by Justin Petrillo | Math Lecture

We will prove below that a system is inter-communicating if and only if it is chaotic. This is the fundamental law of data science, since it is both analytical, as coming from definitions, and synthetic, as explaining empirical observation, thus satisfying the Kantian transcendental deduction for...

## Research Colloquium: COVID-19 NYC-USA Employment Fluctuation Comparative Analysis

by Justin Petrillo | Math Resources

Income Level Employment Comparison Looking at Figure 1, a comparison between the totals for NYC and National, i.e. the blue and purple lines, reveals that NYC employment levels had decreased, relative to January, much more than those nationally. Breaking this down by income level categories,...

## Research Colloquium: The Astro-Socio-Biology of COVID-19

by Justin Petrillo | Math Resources

A multi-scale thermodynamic analysis of COVID-19 is developed on the micro-biological, meso-social, & macro-astrological levels to model the accumulation of errors during processes of self-replication within immune response, communicative functionality within social mitigation strategies, and...

## How to do Data Analysis with Python using SQL – a COVID Econometrics example

by Justin Petrillo | Students Tips

This blog post explains how to do data analysis with Python using SQL. The focus is on connecting to or creating external SQL databases and constructing database frames with Pandas of the relevant information for cleaning, transforming, and other manipulations to the external database. Ultimately...